Question: Let $f(x, y) = x(1 - y)$. Suppose $\vec{a} = (2, 0)$ and $\vec{v} = \left( 0, -1 \right)$. Find the directional derivative of $f(x, y)$ at $\vec{a}$ in the direction of $\vec{v}$. $\dfrac{\partial f}{\partial v} = $
When a directional derivative is in the direction $(1, 0)$, $(0, 1)$, $(-1, 0)$, or $(0, -1)$, it becomes a regular partial derivative. Because $v = (0, -1)$, the directional derivative we want to find is also $-\dfrac{\partial f}{\partial y}$ evaluated at $(2, 0)$. $\begin{aligned} &-\dfrac{\partial f}{\partial y} = x \\ \\ &-\dfrac{\partial f}{\partial y}(2, 0) = 2 \end{aligned}$ In conclusion, the directional derivative of $f$ at $\vec{a}$ in the direction of $\vec{v}$ equals $2$.